• Introduction

     Ariithmetic, assortment} could be a collection of distinct objects, thought of as AN object in its titleas an example, the numbers two, 4, and half-dozen area unit distinct objects once thought of severallyhowever once they area unit thought of conjointly they type one set of size 3, written  . The idea of a collection is one in all the foremost basic in arithmetic. Developed at the tip of the nineteenth century, {set theory|pure arithmetic} is currently a omnipresent a part of mathematics, and may be used as a foundation from that nearly all of arithmetic will be derived. In arithmetic education, elementary topics from pure mathematics like logistician diagrams area unit instructed at a young age, whereas a lot of advanced ideas area unit instructed as a part of a university degree.

    The German word Menge, rendered as "set" in English, was coined by Claude Bernard urban center in his work The Paradoxes of the Infinite.

    A set could be a well-defined assortment of distinct objects. The objects that compose a collection (also called the set's components or members) will be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one in all the founders of pure mathematics, gave the subsequent definition of a collection at the start of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]

    A set could be a gathering along into an entire of definite, distinct objects of our perception [Anschauung] or of our thought—which area unit known as components of the set.

    Sets area unit conventionally denoted with capital letters. Sets A and B area unit equal if and as long as they need exactly constant components.[2]

    For technical reasons, Cantor's definition clad to be inadequate; these days, in contexts wherever a lot of rigor is needed, one will use axiomatic pure mathematicswithin which the notion of a "set" is taken as a primitive notion and also the properties of sets area unit outlined by a set of axioms. the foremost basic properties area unit that a collection will have componentswhich 2 sets area unit equal (one and also the same) if and as long as each component of every set is a component of the other; this property is termed the extensionality of sets

     Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are simply called factors. 

    Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number, say x, greater than 1 is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0. (Again, in such a way that 1 goes to the multiplicand.)

    Another view on multiplication of integer numbers, extendable to rationals, but not very accessible for real numbers, is by considering it as repeated addition. So 3 × 4 corresponds to either adding 3 times a 4, or 4 times a 3, giving the same result. There are different opinions on the advantageousness of these paradigmata in math education.

    Multiplication is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number (no stretching or squeezing). The multiplicative inverse for any number except 0 is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity 10 is the only number without a multiplicative inverse, and the result of multiplying any number and 0 is again 0. One says, 0 is not contained in the multiplicative group of the numbers.

    The product of a and b is written as a × b or a·b. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages in which one can only use characters normally found on a keyboard, it is often written with an asterisk: a * b.

    Algorithms implementing the operation of multiplication for various representations of numbers are by far more costly and laborious than those for addition. Those accessible for manual computation either rely on breaking down the factors to single place values and apply repeated addition, or employ tables or slide rules, thereby mapping the multiplication to addition and back. These methods are outdated and replaced by mobile devices. Computers utilize diverse sophisticated and highly optimized algorithms to implement multiplication and division for the various number formats supported in their system.

    Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal") positional notation; however, any numeral system based on powers of 10, e.g., GreekCyrillicRoman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

    Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).

    The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci[8] using the Hindu–Arabic numeral system.

    Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value 9, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one. This adjustment is termed a carryof the value 1.

    The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds 9, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from 1 to 8 (9 × 9 = 81). Additional steps define the final result.

    Similar techniques exist for subtraction and division.

    The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of 10 increases by 1 (to the left) or decreases by 1 (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10n with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 102, 10, 1, 10−1, 10−2, ...}.

    Repeated multiplication of any value in this list by 10 produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.

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